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Some Convexity Results for the Cartan Decomposition |
Canad. J. Math. 55(2003), 1000-1018
Published:2003-10-01
Printed: Oct 2003
Printed: Oct 2003
- P. Graczyk
- P. Sawyer
Abstract
In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$
where $a(g)$ is the abelian part in the Cartan decomposition of
$g$. This is exactly the support of the measure intervening in the
product formula for the spherical functions on symmetric spaces of
noncompact type. We give a simple description of that support in
the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$,
$\mathbf{C}$ or $\mathbf{H}$. In particular, we show that
$\mathcal{S}$ is convex.
We also give an application of our result to the description of
singular values of a product of two arbitrary matrices with
prescribed singular values.
| Keywords: | convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values |
| MSC Classifications: |
43A90 - Spherical functions [See also 22E45, 22E46, 33C55] 53C35 - Symmetric spaces [See also 32M15, 57T15] 15A18 - Eigenvalues, singular values, and eigenvectors |